A~mixed boundary-value problem for a~hyperbolic-parabolic equation

I. E. Egorov

Matematičeskie zametki, Tome 23 (1978) no. 3, pp. 389-400.

Résumé

Let $\Omega$ be a bounded domain in the $n$-dimensional Euclidean space. In the cylindrical domain $Q_T=\Omega\times[0,T]$ we consider a hyperbolic-parabolic equation of the form $$ Lu=k(x,t)u_{tt}+\sum_{i=1}^na_iu_{tx_i}-\sum_{i,j=1}^n\frac\partial{\partial x_i}(a_{ij}(x,t)u_{x_j})+\sum^n_{i=1}b_iu_{x_i}+au_t+cu=f(x,t),\eqno(1) $$ where $k(x,t)\ge0$, $a_{ij}=a_{ji}$, $\nu|\xi|^2\le a_{ij}\xi_i\xi_j\le\mu|\xi|^2$, $\forall\,\xi\in\mathbf R^n$, $\nu>0$. The classical and the “modified” mixed boundary-value problems for Eq. (1) are studied. Under certain conditions on the coefficients of the equation it is proved that these problems have unique solution in the Sobolev spaces $W_2^1(Q_T)$ и $W_2^2(Q_T)$.